I see this is the expression for the "c" component of the angular momentum but I would like to get the final expression using Einstein's notation and the Levi-Civita, so I guess this could be written as

## i \hbar \epsilon_{cab} x_a p_b ##

Using (3) I get :

## i \hbar L_c ##

Which is different to what I want to prove.

## [L_a,L_b] = i \hbar \epsilon_{abc} L_c ##

Then my assumption is that the Levi Civita symbol in the original equation makes explicit that c is different from a and b and that

## [L_b,L_a] = - [L_a,L_b] ##

So would it be correct to do the following in order to arrive to the solution?

These two expressions are not equivalent. The first expression makes no reference to index c, whereas the second expression does explicitly reference c. Also, the first expression contains fixed values of a and b. But the second expression is a summation over all values of a and b.

In going from ##\sum_u \sum_v \sum_w \epsilon_{vau} \epsilon_{vwb} x_u p_w ## to ##\sum_u \sum_v \sum_w (\delta_{aw}\delta_{ub}-\delta_{ab}\delta_{uw})x_u p_w ##, should there still be a sum over ##v##?

Note, your original work looked good to me all the way to the result of ##i \hbar (x_a p_b- x_b p_a) ## (except for a couple of typos where you had ##x_u x_u## instead of ##x_u p_u## and ##x_t x_t## instead of ##x_t p_t##). The problem was when you said that ##i \hbar (x_a p_b- x_b p_a) = i\hbar \epsilon_{cab} x_a p_b ##.